Equivalent definitions of Clifford algebra, verification

Question

Let $(V,B)$ be a finite dimensional $k$ vector space $V$ with an associated quadratic form $Q$. $char \, k \not= 2$.

Let $X:= \{e_i \}_{i=1}^n$ be a set of basis for $V$. Construct $k\langle X \rangle$, the free associative unital algebra generated by set $X$. Consider the ideal $I:= \langle e_i e_j +e_j e_i + 2B(e_i, e_j) \rangle$,

1. $k\langle X \rangle / I$ satisfies the universal property of the Clifford algebra.

Or equivalently, as the tensor algebra $T(V)$ is free associative algebra of $V$,

2. $T(V)/I'$ satisfies the universal property of the Clifford algebra. $I'$ is defined similarly as $I$.

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What I know: That $Cl(V,Q)$ can be constructed by $T(V)/I''$ where $I'' = \langle v \otimes v - Q(v)\, : \, v \in V \rangle.$

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I believe I could supply proof for both claims if needed. In sketch: Proof for 1. is to use pg 8 of Thomas Friedrich's Dirac Operators; Proof for 2 is to show that $I=I''$.

I just want to know if these claims are true.